Particle filter Tutorial particle filter: From derivation to Application (iii) __ algorithm

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Five, resampling

In the process of applying SIS filtering, there is a problem of degeneration. After several iterations, the weights of many particles become very small, can be ignored, only a small number of particles weight is relatively large. and the variance of particle weights increases with time, and the number of effective particles in state space is less. With the increase of the number of invalid sampling particles, a large amount of computation is wasted on the particle which has little effect on the probability distribution of the estimated posterior filter, which makes the estimation performance decrease, as shown in the figure.

Effective particle numbers are usually used to measure the degradation degree of particle weights, i.e.

The meaning of this formula is that the smaller the number of effective particles, that is, the greater the variance of the weight, that is, the large gap between the weight and the small weights, indicating that the weight degradation is more serious. In practical calculations, the number of effective particles can be approximated by:

In the sequential importance sampling, if the upper formula is smaller than a predetermined threshold, then some measures should be taken to control it. The most direct way to overcome the weight degradation of sequential importance sampling algorithm is to increase the number of particles, which will result in the corresponding increase of computation, and affect the real-time performance of the calculation. Therefore, the following two ways are generally used: (1) Selecting the appropriate importance probability density function, (2) After sequential importance sampling, the Resampling method is used.

For the first method: one criterion for selecting the importance probability density function is to minimize the variance of the particle weight. About this part of the content, or recommended Baidu Library of the article "Particle filter Theory", he also extended here several different particle filtering methods.

The second method, resampling, is highlighted here.

Resampling The idea is: since those small weight does not work, then do not. To keep the number of particles unchanged, some new particles are needed to replace them. The easiest way to find new particles is to copy several of the weight-heavy particles out, as a few. That in the weight of the particles inside let them according to the proportion of their weight to allocate, that is, the eldest brother is divided into the most, the second more points, and so on. The following is illustrated in mathematical form.

It has been explained earlier that the question of seeking a certain expectation becomes the form of this weighted sum:

(1)

After resampling, you want to represent:

(2)

, pay attention to contrast (1) and (2). It's the K-time particle. is the particle after the K-time resampling. where n (i) is the number of times a particle is copied when it produces a new set of particles. (2) The first equal sign in the formula shows that after resampling, all the particle weights are 1/n, but there are more N (i) times in some particles.

The idea has, depends on the concrete operation method. In the paper of the resampling algorithms for particle fi lters, four methods of resampling are discussed. These four methods are very similar to each other. If you have access to genetic algorithms, it's easy to understand, which is the idea of roulette in genetic algorithms.

Here is a simple example to illustrate:

Suppose there are 3 particles, at the time of the K, their weights are 0.1, 0.1, 0.8, and then calculate their probability accumulation and (in MATLAB for Cumsum ()) Get: [0.1, 0.2, 1]. We then randomly sampled 3 values with a uniform distribution between [0,1], assuming 0.15, 0.38, and 0.54. In other words, the second particle is replicated once, and the third particle is copied two times.

In MATLAB, a command can facilitate the implementation of this process:

[~, J] = HISTC (rand (n,1), [0 cumsum (w ')]); The use of HISTC can be clicked on HISTC usage.

For the above procedure, you can also deepen your understanding of the following diagram:

By putting the Resampling method into the previous SIS algorithm, the elementary particle filtering algorithm is formed.

The idea of resampling is simple, but when you carefully analyze the calculation formula for weights:

There will be questions, the weight of the large number of copies more than a few times, this must be correct. Weight, if the molecular large, that is, after the probability of large, it is true that the probability of a large number of places to put more particles. But the weight is also likely to be a small denominator, this time the molecule may also be small, that is, the actual posterior probability may also be small, this time the big weight may not be so excellent. Moreover, this simple resampling will cause the particle diversity to be lost, and in the end it may turn into a single particle. In genetic algorithm, mutation is also introduced to solve the problem of diversity. Of course, the particle filter also has a special method: regular particle filter, interested in access to relevant information.

At this point, the whole process of particle filtering has been clearly clear, in practical applications there are some uncertainties is the importance of the probability density of choice. In the next chapter, we first introduce the Sir Particle filter, and then use the Sir Filter for practical application.

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Reference

1. J. Gordon "Beyond The Filter:particle filters for tracking Applications"

2. Baidu Library "Particle Filter Theory"

3. Jeroen D. Hol "on resampling algorithms for particle fi lters"

4. Particle filters:how to do resampling?

5. Gabriel A. Terejanu "Tutorial on Monte Carlo techniques"